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   "source": [
    "using VMLS\n",
    "using LinearAlgebra"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Chapter 13\n",
    "# Least squares data fitting\n",
    "### 13.1 Least squares data fitting\n",
    "**Straight-line fit.** A straight-line fit to time series data gives an estimate of a\n",
    "trend line. In Figure [13.3](https://web.stanford.edu/~boyd/vmls/vmls.pdf#figure.13.3) of VMLS we apply this to a time series of petroleum consumption. The figure is reproduced here as Figure 13.1."
   ]
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  {
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   "source": [
    "# Petroleum consumption in thousand barrels/day\n",
    "consumption = petroleum_consumption_data()\n",
    "n = length(consumption);\n",
    "A = [ ones(n) 1:n ];\n",
    "x = A \\ consumption;\n",
    "using Plots\n",
    "scatter(1980:2013, consumption, legend=false)\n",
    "plot!(1980:2013, A*x)"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "**Figure 13.1** World petroleum consumption between $1980$ and $2013$ (dots)\n",
    "and least squares straight-line fit (data from www.eia.gov).\n",
    "\n",
    "**Estimation of trend and seasonal component.** The next example is the least squares fit of a trend plus a periodic component to a time series. In VMLS this was illustrated with a time series of vehicle miles traveled in the US, per month, for $15$ years ($2000–2014$). The following Julia code replicates Figure [13.5](https://web.stanford.edu/~boyd/vmls/vmls.pdf#figure.13.5) in VMLS. It imports the data via the function `vehicle_miles_data`, which creates a $15 × 12$ matrix `vmt`, with the monthly values for each of the $15$ years."
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     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    " vmt = vehicle_miles_data(); # creates 15x12 matrix vmt\n",
    " m = 15*12;\n",
    " A = [ 0:(m-1) vcat([eye(12) for i=1:15]...) ];\n",
    " b = reshape(vmt', m, 1);\n",
    " x = A \\ b;\n",
    " using Plots\n",
    " scatter(1:m, b, markersize = 2, legend =false);\n",
    " plot!(1:m, A*x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='page_100'></a>\n",
    "**Figure 13.2** The dots show vehicle miles traveled in the US, per month, in the period January $2000$–December $2014$. The line shows the least squares fit of a linear trend and a seasonal component with a $12$-month period.\n",
    "\n",
    "The matrix $A$ in this example has size $m×n$ where $m = 15 · 12 = 180$ and $n = 13$. The first column has entries $0, 1, 2, . . . , 179$. The remaining columns are formed by vertical stacking of $15$ identity matrices of size $12 × 12$. The Julia expression `vcat([eye(12) for i=1:15]...)` creates an array of 15 identity matrices, and then stacks them vertically. The plot produced by the code is shown in Figure 13.2. \n",
    "\n",
    "**Polynomial fit.** We now discuss the polynomial fitting problem on page [255](        https://web.stanford.edu/~boyd/vmls/vmls.pdf#section*.300) in VMLS and the results shown in Figure [13.6](https://web.stanford.edu/~boyd/vmls/vmls.pdf#figure.13.6). We first generate a training set of $100$ points and plot them (Figure 13.3)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Generate training data in the interval [-1, 1].\n",
    "m = 100;\n",
    "t = -1 .+ 2*rand(m,1);\n",
    "y = t.^3 - t + 0.4 ./ (1 .+ 25*t.^2) + 0.10*randn(m,1);\n",
    "using Plots\n",
    "scatter(t,y,legend=false)"
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Figure 13.3** Training set used in the polynomial fitting example.\n",
    "\n",
    "Next we define a function that fits the polynomial coefficients using least squares.\n",
    "We apply the function to fit polynomials of degree $2, 6, 10, 15$ to our training set."
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  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
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       "3×1 Array{Float64,2}:\n",
       "  0.21000591069637362\n",
       " -0.3961556355504553 \n",
       " -0.23191808988500345"
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   "source": [
    "polyfit(t, y, p) = vandermonde(t, p) \\ y\n",
    "theta2 = polyfit(t,y,3)"
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  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "theta6 = polyfit(t,y,7);\n",
    "theta10 = polyfit(t,y,11);\n",
    "theta15 = polyfit(t,y,16);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we plot the four polynomials. To simplify this, we first write a function that evaluates a polynomial at all points specified in a vector `x`. The plots are in Figure 13.4."
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   "cell_type": "code",
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       "  1694.01,165.67 1694.9,165.589 1695.79,165.509 1696.68,165.43 1697.57,165.353 1698.46,165.278 1699.35,165.204 1700.25,165.132 1701.14,165.061 1702.03,164.992 \n",
       "  1702.92,164.924 1703.81,164.858 1704.7,164.794 1705.59,164.732 1706.49,164.672 1707.38,164.613 1708.27,164.556 1709.16,164.501 1710.05,164.448 1710.94,164.397 \n",
       "  1711.83,164.348 1712.73,164.301 1713.62,164.256 1714.51,164.213 1715.4,164.172 1716.29,164.134 1717.18,164.097 1718.07,164.063 1718.97,164.031 1719.86,164.002 \n",
       "  1720.75,163.975 1721.64,163.95 1722.53,163.927 1723.42,163.907 1724.31,163.89 1725.21,163.875 1726.1,163.862 1726.99,163.852 1727.88,163.845 1728.77,163.841 \n",
       "  1729.66,163.839 1730.55,163.84 1731.45,163.843 1732.34,163.85 1733.23,163.859 1734.12,163.871 1735.01,163.887 1735.9,163.905 1736.79,163.926 1737.69,163.95 \n",
       "  1738.58,163.977 1739.47,164.008 1740.36,164.041 1741.25,164.078 1742.14,164.118 1743.04,164.161 1743.93,164.207 1744.82,164.257 1745.71,164.31 1746.6,164.367 \n",
       "  1747.49,164.427 1748.38,164.49 1749.28,164.557 1750.17,164.628 1751.06,164.702 1751.95,164.78 1752.84,164.861 1753.73,164.946 1754.62,165.035 1755.52,165.128 \n",
       "  1756.41,165.224 1757.3,165.325 1758.19,165.429 1759.08,165.537 1759.97,165.649 1760.86,165.765 1761.76,165.885 1762.65,166.009 1763.54,166.138 1764.43,166.27 \n",
       "  1765.32,166.407 1766.21,166.547 1767.1,166.692 1768,166.842 1768.89,166.995 1769.78,167.153 1770.67,167.315 1771.56,167.482 1772.45,167.653 1773.34,167.829 \n",
       "  1774.24,168.009 1775.13,168.194 1776.02,168.383 1776.91,168.577 1777.8,168.775 1778.69,168.978 1779.58,169.186 1780.48,169.398 1781.37,169.616 1782.26,169.838 \n",
       "  1783.15,170.065 1784.04,170.297 1784.93,170.533 1785.82,170.775 1786.72,171.021 1787.61,171.273 1788.5,171.529 1789.39,171.791 1790.28,172.057 1791.17,172.329 \n",
       "  1792.06,172.606 1792.96,172.888 1793.85,173.175 1794.74,173.467 1795.63,173.765 1796.52,174.068 1797.41,174.376 1798.3,174.689 1799.2,175.008 1800.09,175.332 \n",
       "  1800.98,175.662 1801.87,175.996 1802.76,176.337 1803.65,176.682 1804.54,177.034 1805.44,177.39 1806.33,177.753 1807.22,178.121 1808.11,178.494 1809,178.873 \n",
       "  1809.89,179.258 1810.78,179.648 1811.68,180.044 1812.57,180.445 1813.46,180.853 1814.35,181.266 1815.24,181.684 1816.13,182.109 1817.02,182.539 1817.92,182.975 \n",
       "  1818.81,183.417 1819.7,183.865 1820.59,184.318 1821.48,184.778 1822.37,185.243 1823.27,185.714 1824.16,186.191 1825.05,186.674 1825.94,187.163 1826.83,187.658 \n",
       "  1827.72,188.159 1828.61,188.666 1829.51,189.178 1830.4,189.697 1831.29,190.222 1832.18,190.753 1833.07,191.29 1833.96,191.833 1834.85,192.382 1835.75,192.937 \n",
       "  1836.64,193.498 1837.53,194.065 1838.42,194.639 1839.31,195.218 1840.2,195.804 1841.09,196.396 1841.99,196.993 1842.88,197.597 1843.77,198.207 1844.66,198.824 \n",
       "  1845.55,199.446 1846.44,200.074 1847.33,200.709 1848.23,201.35 1849.12,201.997 1850.01,202.65 1850.9,203.31 1851.79,203.975 1852.68,204.647 1853.57,205.325 \n",
       "  1854.47,206.009 1855.36,206.699 1856.25,207.395 1857.14,208.098 1858.03,208.806 1858.92,209.521 1859.81,210.242 1860.71,210.969 1861.6,211.703 1862.49,212.442 \n",
       "  1863.38,213.188 1864.27,213.939 1865.16,214.697 1866.05,215.461 1866.95,216.231 1867.84,217.007 1868.73,217.79 1869.62,218.578 1870.51,219.373 1871.4,220.173 \n",
       "  1872.29,220.98 1873.19,221.793 1874.08,222.611 1874.97,223.436 1875.86,224.267 1876.75,225.104 1877.64,225.946 1878.53,226.795 1879.43,227.65 1880.32,228.511 \n",
       "  1881.21,229.377 1882.1,230.25 1882.99,231.128 1883.88,232.013 1884.77,232.903 1885.67,233.799 1886.56,234.701 1887.45,235.609 1888.34,236.522 1889.23,237.441 \n",
       "  1890.12,238.367 1891.01,239.297 1891.91,240.234 1892.8,241.176 1893.69,242.124 1894.58,243.077 1895.47,244.037 1896.36,245.001 1897.25,245.972 1898.15,246.948 \n",
       "  1899.04,247.929 1899.93,248.916 1900.82,249.908 1901.71,250.906 1902.6,251.909 1903.49,252.918 1904.39,253.932 1905.28,254.951 1906.17,255.976 1907.06,257.006 \n",
       "  1907.95,258.041 1908.84,259.082 1909.74,260.127 1910.63,261.178 1911.52,262.234 1912.41,263.295 1913.3,264.361 1914.19,265.432 1915.08,266.507 1915.98,267.588 \n",
       "  1916.87,268.674 1917.76,269.765 1918.65,270.86 1919.54,271.961 1920.43,273.066 1921.32,274.175 1922.22,275.29 1923.11,276.409 1924,277.533 1924.89,278.661 \n",
       "  1925.78,279.794 1926.67,280.931 1927.56,282.073 1928.46,283.219 1929.35,284.369 1930.24,285.524 1931.13,286.683 1932.02,287.847 1932.91,289.014 1933.8,290.186 \n",
       "  1934.7,291.361 1935.59,292.541 1936.48,293.725 1937.37,294.912 1938.26,296.104 1939.15,297.299 1940.04,298.498 1940.94,299.701 1941.83,300.908 1942.72,302.118 \n",
       "  1943.61,303.332 1944.5,304.55 1945.39,305.771 1946.28,306.995 1947.18,308.223 1948.07,309.454 1948.96,310.688 1949.85,311.926 1950.74,313.167 1951.63,314.411 \n",
       "  1952.52,315.658 1953.42,316.908 1954.31,318.161 1955.2,319.417 1956.09,320.676 1956.98,321.937 1957.87,323.202 1958.76,324.468 1959.66,325.738 1960.55,327.01 \n",
       "  1961.44,328.285 1962.33,329.562 1963.22,330.841 1964.11,332.123 1965,333.407 1965.9,334.693 1966.79,335.981 1967.68,337.272 1968.57,338.564 1969.46,339.858 \n",
       "  1970.35,341.155 1971.24,342.453 1972.14,343.752 1973.03,345.054 1973.92,346.357 1974.81,347.661 1975.7,348.968 1976.59,350.275 1977.48,351.584 1978.38,352.894 \n",
       "  1979.27,354.205 1980.16,355.518 1981.05,356.831 1981.94,358.146 1982.83,359.462 1983.72,360.778 1984.62,362.095 1985.51,363.413 1986.4,364.732 1987.29,366.051 \n",
       "  1988.18,367.371 1989.07,368.691 1989.97,370.012 1990.86,371.332 1991.75,372.654 1992.64,373.975 1993.53,375.296 1994.42,376.618 1995.31,377.939 1996.21,379.26 \n",
       "  1997.1,380.581 1997.99,381.902 1998.88,383.222 1999.77,384.542 2000.66,385.862 2001.55,387.181 2002.45,388.499 2003.34,389.816 2004.23,391.133 2005.12,392.449 \n",
       "  2006.01,393.763 2006.9,395.077 2007.79,396.39 2008.69,397.701 2009.58,399.012 2010.47,400.321 2011.36,401.628 2012.25,402.934 2013.14,404.238 2014.03,405.541 \n",
       "  2014.93,406.842 2015.82,408.141 2016.71,409.438 2017.6,410.733 2018.49,412.027 2019.38,413.318 2020.27,414.606 2021.17,415.893 2022.06,417.177 2022.95,418.459 \n",
       "  2023.84,419.738 2024.73,421.014 2025.62,422.288 2026.51,423.559 2027.41,424.827 2028.3,426.093 2029.19,427.355 2030.08,428.614 2030.97,429.87 2031.86,431.122 \n",
       "  2032.75,432.371 2033.65,433.617 2034.54,434.859 2035.43,436.098 2036.32,437.333 2037.21,438.564 2038.1,439.791 2038.99,441.015 2039.89,442.234 2040.78,443.449 \n",
       "  2041.67,444.66 2042.56,445.867 2043.45,447.069 2044.34,448.267 2045.23,449.461 2046.13,450.65 2047.02,451.834 2047.91,453.013 2048.8,454.188 2049.69,455.357 \n",
       "  2050.58,456.522 2051.47,457.681 2052.37,458.836 2053.26,459.985 2054.15,461.128 2055.04,462.266 2055.93,463.399 2056.82,464.526 2057.71,465.647 2058.61,466.763 \n",
       "  2059.5,467.873 2060.39,468.977 2061.28,470.074 2062.17,471.166 2063.06,472.251 2063.95,473.331 2064.85,474.404 2065.74,475.47 2066.63,476.53 2067.52,477.583 \n",
       "  2068.41,478.63 2069.3,479.67 2070.19,480.703 2071.09,481.729 2071.98,482.748 2072.87,483.76 2073.76,484.765 2074.65,485.762 2075.54,486.753 2076.44,487.735 \n",
       "  2077.33,488.711 2078.22,489.679 2079.11,490.639 2080,491.591 2080.89,492.536 2081.78,493.472 2082.68,494.401 2083.57,495.322 2084.46,496.234 2085.35,497.139 \n",
       "  2086.24,498.035 2087.13,498.922 2088.02,499.802 2088.92,500.672 2089.81,501.535 2090.7,502.388 2091.59,503.233 2092.48,504.069 2093.37,504.896 2094.26,505.714 \n",
       "  2095.16,506.523 2096.05,507.323 2096.94,508.113 2097.83,508.895 2098.72,509.667 2099.61,510.429 2100.5,511.183 2101.4,511.926 2102.29,512.66 2103.18,513.384 \n",
       "  2104.07,514.099 2104.96,514.804 2105.85,515.498 2106.74,516.183 2107.64,516.858 2108.53,517.522 2109.42,518.177 2110.31,518.821 2111.2,519.455 2112.09,520.078 \n",
       "  2112.98,520.691 2113.88,521.293 2114.77,521.885 2115.66,522.466 2116.55,523.037 2117.44,523.597 2118.33,524.145 2119.22,524.683 2120.12,525.21 2121.01,525.726 \n",
       "  2121.9,526.231 2122.79,526.725 2123.68,527.207 2124.57,527.679 2125.46,528.138 2126.36,528.587 2127.25,529.024 2128.14,529.45 2129.03,529.864 2129.92,530.266 \n",
       "  2130.81,530.657 2131.7,531.036 2132.6,531.403 2133.49,531.759 2134.38,532.102 2135.27,532.434 2136.16,532.753 2137.05,533.061 2137.94,533.357 2138.84,533.64 \n",
       "  2139.73,533.911 2140.62,534.17 2141.51,534.417 2142.4,534.652 2143.29,534.874 2144.18,535.084 2145.08,535.281 2145.97,535.466 2146.86,535.638 2147.75,535.798 \n",
       "  2148.64,535.945 2149.53,536.08 2150.42,536.202 2151.32,536.311 2152.21,536.408 2153.1,536.491 2153.99,536.562 2154.88,536.62 2155.77,536.666 2156.67,536.698 \n",
       "  2157.56,536.717 2158.45,536.724 2159.34,536.717 2160.23,536.698 2161.12,536.665 2162.01,536.62 2162.91,536.561 2163.8,536.489 2164.69,536.404 2165.58,536.306 \n",
       "  2166.47,536.195 2167.36,536.071 2168.25,535.933 2169.15,535.782 2170.04,535.618 2170.93,535.441 2171.82,535.25 2172.71,535.047 2173.6,534.829 2174.49,534.599 \n",
       "  2175.39,534.355 2176.28,534.098 2177.17,533.828 2178.06,533.545 2178.95,533.248 2179.84,532.938 2180.73,532.614 2181.63,532.277 2182.52,531.927 2183.41,531.564 \n",
       "  2184.3,531.187 2185.19,530.797 2186.08,530.394 2186.97,529.977 2187.87,529.547 2188.76,529.104 2189.65,528.648 2190.54,528.178 2191.43,527.696 2192.32,527.2 \n",
       "  2193.21,526.69 2194.11,526.168 2195,525.633 2195.89,525.084 2196.78,524.522 2197.67,523.948 2198.56,523.36 2199.45,522.759 2200.35,522.145 2201.24,521.519 \n",
       "  2202.13,520.879 2203.02,520.226 2203.91,519.561 2204.8,518.883 2205.69,518.192 2206.59,517.488 2207.48,516.772 2208.37,516.043 2209.26,515.302 2210.15,514.547 \n",
       "  2211.04,513.781 2211.93,513.002 2212.83,512.21 2213.72,511.406 2214.61,510.59 2215.5,509.762 2216.39,508.922 2217.28,508.069 2218.17,507.204 2219.07,506.328 \n",
       "  2219.96,505.439 2220.85,504.539 2221.74,503.627 2222.63,502.703 2223.52,501.767 2224.41,500.82 2225.31,499.862 2226.2,498.892 2227.09,497.91 2227.98,496.918 \n",
       "  2228.87,495.914 2229.76,494.9 2230.65,493.874 2231.55,492.837 2232.44,491.79 2233.33,490.732 2234.22,489.663 2235.11,488.584 2236,487.494 2236.89,486.394 \n",
       "  2237.79,485.284 2238.68,484.164 2239.57,483.034 2240.46,481.894 2241.35,480.744 2242.24,479.584 2243.14,478.415 2244.03,477.237 2244.92,476.049 2245.81,474.852 \n",
       "  2246.7,473.646 2247.59,472.431 2248.48,471.208 2249.38,469.975 2250.27,468.734 2251.16,467.485 2252.05,466.228 2252.94,464.962 2253.83,463.688 2254.72,462.407 \n",
       "  2255.62,461.118 2256.51,459.821 2257.4,458.517 2258.29,457.206 2259.18,455.888 2260.07,454.563 2260.96,453.231 2261.86,451.893 2262.75,450.548 2263.64,449.197 \n",
       "  2264.53,447.84 2265.42,446.477 2266.31,445.108 2267.2,443.733 2268.1,442.354 2268.99,440.969 2269.88,439.579 2270.77,438.185 2271.66,436.785 2272.55,435.382 \n",
       "  2273.44,433.974 2274.34,432.562 2275.23,431.146 2276.12,429.727 2277.01,428.304 2277.9,426.878 2278.79,425.45 2279.68,424.018 2280.58,422.584 2281.47,421.147 \n",
       "  2282.36,419.709 2283.25,418.268 2284.14,416.826 2285.03,415.382 2285.92,413.938 2286.82,412.492 2287.71,411.045 2288.6,409.599 2289.49,408.152 2290.38,406.704 \n",
       "  2291.27,405.258 2292.16,403.811 2293.06,402.366 2293.95,400.922 2294.84,399.478 2295.73,398.037 2296.62,396.597 2297.51,395.16 2298.4,393.725 2299.3,392.292 \n",
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       "  1827.72,979.173 1828.61,979.186 1829.51,979.209 1830.4,979.243 1831.29,979.288 1832.18,979.343 1833.07,979.41 1833.96,979.487 1834.85,979.576 1835.75,979.677 \n",
       "  1836.64,979.789 1837.53,979.913 1838.42,980.048 1839.31,980.196 1840.2,980.356 1841.09,980.529 1841.99,980.714 1842.88,980.911 1843.77,981.121 1844.66,981.344 \n",
       "  1845.55,981.58 1846.44,981.828 1847.33,982.09 1848.23,982.365 1849.12,982.653 1850.01,982.955 1850.9,983.27 1851.79,983.599 1852.68,983.942 1853.57,984.298 \n",
       "  1854.47,984.668 1855.36,985.052 1856.25,985.45 1857.14,985.862 1858.03,986.288 1858.92,986.728 1859.81,987.183 1860.71,987.652 1861.6,988.135 1862.49,988.633 \n",
       "  1863.38,989.145 1864.27,989.671 1865.16,990.212 1866.05,990.768 1866.95,991.338 1867.84,991.923 1868.73,992.523 1869.62,993.137 1870.51,993.767 1871.4,994.411 \n",
       "  1872.29,995.069 1873.19,995.743 1874.08,996.431 1874.97,997.135 1875.86,997.853 1876.75,998.586 1877.64,999.334 1878.53,1000.1 1879.43,1000.87 1880.32,1001.67 \n",
       "  1881.21,1002.48 1882.1,1003.3 1882.99,1004.14 1883.88,1004.99 1884.77,1005.86 1885.67,1006.74 1886.56,1007.63 1887.45,1008.55 1888.34,1009.47 1889.23,1010.42 \n",
       "  1890.12,1011.37 1891.01,1012.34 1891.91,1013.33 1892.8,1014.33 1893.69,1015.35 1894.58,1016.38 1895.47,1017.42 1896.36,1018.48 1897.25,1019.55 1898.15,1020.64 \n",
       "  1899.04,1021.75 1899.93,1022.86 1900.82,1023.99 1901.71,1025.14 1902.6,1026.3 1903.49,1027.48 1904.39,1028.66 1905.28,1029.87 1906.17,1031.09 1907.06,1032.32 \n",
       "  1907.95,1033.56 1908.84,1034.82 1909.74,1036.09 1910.63,1037.38 1911.52,1038.68 1912.41,1040 1913.3,1041.32 1914.19,1042.66 1915.08,1044.02 1915.98,1045.39 \n",
       "  1916.87,1046.77 1917.76,1048.16 1918.65,1049.57 1919.54,1050.99 1920.43,1052.42 1921.32,1053.87 1922.22,1055.33 1923.11,1056.8 1924,1058.28 1924.89,1059.78 \n",
       "  1925.78,1061.29 1926.67,1062.81 1927.56,1064.34 1928.46,1065.88 1929.35,1067.44 1930.24,1069.01 1931.13,1070.59 1932.02,1072.18 1932.91,1073.78 1933.8,1075.39 \n",
       "  1934.7,1077.02 1935.59,1078.65 1936.48,1080.3 1937.37,1081.95 1938.26,1083.62 1939.15,1085.3 1940.04,1086.98 1940.94,1088.68 1941.83,1090.39 1942.72,1092.1 \n",
       "  1943.61,1093.83 1944.5,1095.56 1945.39,1097.31 1946.28,1099.06 1947.18,1100.82 1948.07,1102.59 1948.96,1104.37 1949.85,1106.16 1950.74,1107.95 1951.63,1109.76 \n",
       "  1952.52,1111.57 1953.42,1113.38 1954.31,1115.21 1955.2,1117.04 1956.09,1118.88 1956.98,1120.72 1957.87,1122.57 1958.76,1124.43 1959.66,1126.29 1960.55,1128.16 \n",
       "  1961.44,1130.04 1962.33,1131.91 1963.22,1133.8 1964.11,1135.69 1965,1137.58 1965.9,1139.47 1966.79,1141.37 1967.68,1143.28 1968.57,1145.18 1969.46,1147.09 \n",
       "  1970.35,1149.01 1971.24,1150.92 1972.14,1152.84 1973.03,1154.75 1973.92,1156.67 1974.81,1158.6 1975.7,1160.52 1976.59,1162.44 1977.48,1164.36 1978.38,1166.28 \n",
       "  1979.27,1168.21 1980.16,1170.13 1981.05,1172.05 1981.94,1173.97 1982.83,1175.89 1983.72,1177.8 1984.62,1179.72 1985.51,1181.63 1986.4,1183.53 1987.29,1185.44 \n",
       "  1988.18,1187.34 1989.07,1189.24 1989.97,1191.13 1990.86,1193.01 1991.75,1194.9 1992.64,1196.77 1993.53,1198.65 1994.42,1200.51 1995.31,1202.37 1996.21,1204.22 \n",
       "  1997.1,1206.06 1997.99,1207.9 1998.88,1209.73 1999.77,1211.55 2000.66,1213.36 2001.55,1215.17 2002.45,1216.96 2003.34,1218.74 2004.23,1220.52 2005.12,1222.28 \n",
       "  2006.01,1224.03 2006.9,1225.77 2007.79,1227.5 2008.69,1229.22 2009.58,1230.92 2010.47,1232.61 2011.36,1234.29 2012.25,1235.96 2013.14,1237.61 2014.03,1239.24 \n",
       "  2014.93,1240.87 2015.82,1242.48 2016.71,1244.07 2017.6,1245.64 2018.49,1247.21 2019.38,1248.75 2020.27,1250.28 2021.17,1251.79 2022.06,1253.28 2022.95,1254.76 \n",
       "  2023.84,1256.22 2024.73,1257.66 2025.62,1259.08 2026.51,1260.48 2027.41,1261.86 2028.3,1263.23 2029.19,1264.57 2030.08,1265.89 2030.97,1267.2 2031.86,1268.48 \n",
       "  2032.75,1269.74 2033.65,1270.98 2034.54,1272.2 2035.43,1273.4 2036.32,1274.58 2037.21,1275.73 2038.1,1276.86 2038.99,1277.97 2039.89,1279.05 2040.78,1280.12 \n",
       "  2041.67,1281.16 2042.56,1282.17 2043.45,1283.17 2044.34,1284.13 2045.23,1285.08 2046.13,1286 2047.02,1286.9 2047.91,1287.77 2048.8,1288.62 2049.69,1289.44 \n",
       "  2050.58,1290.24 2051.47,1291.01 2052.37,1291.76 2053.26,1292.49 2054.15,1293.19 2055.04,1293.86 2055.93,1294.51 2056.82,1295.14 2057.71,1295.74 2058.61,1296.31 \n",
       "  2059.5,1296.86 2060.39,1297.39 2061.28,1297.89 2062.17,1298.36 2063.06,1298.81 2063.95,1299.24 2064.85,1299.64 2065.74,1300.02 2066.63,1300.37 2067.52,1300.7 \n",
       "  2068.41,1301.01 2069.3,1301.29 2070.19,1301.55 2071.09,1301.78 2071.98,1301.99 2072.87,1302.18 2073.76,1302.35 2074.65,1302.49 2075.54,1302.61 2076.44,1302.71 \n",
       "  2077.33,1302.79 2078.22,1302.84 2079.11,1302.88 2080,1302.89 2080.89,1302.89 2081.78,1302.86 2082.68,1302.81 2083.57,1302.75 2084.46,1302.67 2085.35,1302.56 \n",
       "  2086.24,1302.44 2087.13,1302.31 2088.02,1302.15 2088.92,1301.98 2089.81,1301.79 2090.7,1301.59 2091.59,1301.37 2092.48,1301.14 2093.37,1300.9 2094.26,1300.64 \n",
       "  2095.16,1300.36 2096.05,1300.08 2096.94,1299.78 2097.83,1299.47 2098.72,1299.15 2099.61,1298.82 2100.5,1298.48 2101.4,1298.14 2102.29,1297.78 2103.18,1297.42 \n",
       "  2104.07,1297.05 2104.96,1296.67 2105.85,1296.29 2106.74,1295.9 2107.64,1295.51 2108.53,1295.11 2109.42,1294.71 2110.31,1294.31 2111.2,1293.91 2112.09,1293.5 \n",
       "  2112.98,1293.1 2113.88,1292.7 2114.77,1292.29 2115.66,1291.89 2116.55,1291.49 2117.44,1291.1 2118.33,1290.71 2119.22,1290.32 2120.12,1289.94 2121.01,1289.57 \n",
       "  2121.9,1289.2 2122.79,1288.84 2123.68,1288.48 2124.57,1288.14 2125.46,1287.8 2126.36,1287.48 2127.25,1287.16 2128.14,1286.86 2129.03,1286.57 2129.92,1286.29 \n",
       "  2130.81,1286.02 2131.7,1285.77 2132.6,1285.53 2133.49,1285.31 2134.38,1285.1 2135.27,1284.91 2136.16,1284.74 2137.05,1284.58 2137.94,1284.44 2138.84,1284.32 \n",
       "  2139.73,1284.22 2140.62,1284.14 2141.51,1284.07 2142.4,1284.03 2143.29,1284.01 2144.18,1284.01 2145.08,1284.03 2145.97,1284.07 2146.86,1284.13 2147.75,1284.21 \n",
       "  2148.64,1284.32 2149.53,1284.45 2150.42,1284.61 2151.32,1284.78 2152.21,1284.98 2153.1,1285.2 2153.99,1285.45 2154.88,1285.72 2155.77,1286.02 2156.67,1286.33 \n",
       "  2157.56,1286.68 2158.45,1287.04 2159.34,1287.43 2160.23,1287.84 2161.12,1288.28 2162.01,1288.74 2162.91,1289.22 2163.8,1289.73 2164.69,1290.26 2165.58,1290.81 \n",
       "  2166.47,1291.38 2167.36,1291.98 2168.25,1292.59 2169.15,1293.23 2170.04,1293.89 2170.93,1294.56 2171.82,1295.26 2172.71,1295.98 2173.6,1296.71 2174.49,1297.46 \n",
       "  2175.39,1298.23 2176.28,1299.02 2177.17,1299.82 2178.06,1300.63 2178.95,1301.46 2179.84,1302.3 2180.73,1303.15 2181.63,1304.02 2182.52,1304.89 2183.41,1305.78 \n",
       "  2184.3,1306.67 2185.19,1307.57 2186.08,1308.48 2186.97,1309.39 2187.87,1310.3 2188.76,1311.22 2189.65,1312.14 2190.54,1313.06 2191.43,1313.98 2192.32,1314.89 \n",
       "  2193.21,1315.81 2194.11,1316.71 2195,1317.61 2195.89,1318.51 2196.78,1319.39 2197.67,1320.26 2198.56,1321.13 2199.45,1321.97 2200.35,1322.81 2201.24,1323.62 \n",
       "  2202.13,1324.42 2203.02,1325.2 2203.91,1325.96 2204.8,1326.69 2205.69,1327.4 2206.59,1328.09 2207.48,1328.75 2208.37,1329.38 2209.26,1329.98 2210.15,1330.54 \n",
       "  2211.04,1331.08 2211.93,1331.58 2212.83,1332.04 2213.72,1332.47 2214.61,1332.85 2215.5,1333.2 2216.39,1333.5 2217.28,1333.76 2218.17,1333.97 2219.07,1334.14 \n",
       "  2219.96,1334.26 2220.85,1334.33 2221.74,1334.34 2222.63,1334.31 2223.52,1334.23 2224.41,1334.09 2225.31,1333.89 2226.2,1333.64 2227.09,1333.33 2227.98,1332.97 \n",
       "  2228.87,1332.54 2229.76,1332.05 2230.65,1331.51 2231.55,1330.9 2232.44,1330.23 2233.33,1329.5 2234.22,1328.7 2235.11,1327.84 2236,1326.92 2236.89,1325.93 \n",
       "  2237.79,1324.88 2238.68,1323.76 2239.57,1322.58 2240.46,1321.33 2241.35,1320.02 2242.24,1318.65 2243.14,1317.21 2244.03,1315.71 2244.92,1314.15 2245.81,1312.53 \n",
       "  2246.7,1310.84 2247.59,1309.1 2248.48,1307.29 2249.38,1305.43 2250.27,1303.51 2251.16,1301.54 2252.05,1299.51 2252.94,1297.43 2253.83,1295.3 2254.72,1293.12 \n",
       "  2255.62,1290.9 2256.51,1288.63 2257.4,1286.32 2258.29,1283.96 2259.18,1281.57 2260.07,1279.15 2260.96,1276.69 2261.86,1274.2 2262.75,1271.68 2263.64,1269.14 \n",
       "  2264.53,1266.58 2265.42,1264 2266.31,1261.4 2267.2,1258.8 2268.1,1256.18 2268.99,1253.56 2269.88,1250.94 2270.77,1248.32 2271.66,1245.7 2272.55,1243.1 \n",
       "  2273.44,1240.5 2274.34,1237.93 2275.23,1235.37 2276.12,1232.84 2277.01,1230.33 2277.9,1227.86 2278.79,1225.42 2279.68,1223.02 2280.58,1220.66 2281.47,1218.34 \n",
       "  2282.36,1216.07 2283.25,1213.86 2284.14,1211.7 2285.03,1209.6 2285.92,1207.55 2286.82,1205.57 2287.71,1203.66 2288.6,1201.81 2289.49,1200.03 2290.38,1198.32 \n",
       "  2291.27,1196.68 2292.16,1195.12 2293.06,1193.62 2293.95,1192.2 2294.84,1190.85 2295.73,1189.57 2296.62,1188.36 2297.51,1187.22 2298.4,1186.14 2299.3,1185.12 \n",
       "  2300.19,1184.16 2301.08,1183.24 2301.97,1182.38 2302.86,1181.55 2303.75,1180.75 2304.64,1179.98 2305.54,1179.22 2306.43,1178.47 2307.32,1177.7 2308.21,1176.91 \n",
       "  2309.1,1176.09 2309.99,1175.21 2310.88,1174.27 2311.78,1173.24 2312.67,1172.1 2313.56,1170.83 2314.45,1169.41 2315.34,1167.81 2316.23,1166 2317.12,1163.96 \n",
       "  2318.02,1161.66 2318.91,1159.05 2319.8,1156.11 2320.69,1152.79 2321.58,1149.06 2322.47,1144.86 2323.37,1140.16 2324.26,1134.91 2325.15,1129.05 2326.04,1122.53 \n",
       "  \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "polyeval(theta, x) = vandermonde(x,length(theta))*theta;\n",
    "t_plot = linspace(-1,1,1000);\n",
    "using Plots\n",
    "p = plot(layout=4, legend=false, ylim=(-0.7, 0.7))\n",
    "scatter!(t, y, subplot=1, markersize = 2)\n",
    "plot!(t_plot, polyeval(theta2,t_plot), subplot=1)\n",
    "scatter!(t, y, subplot=2, markersize = 2)\n",
    "plot!(t_plot, polyeval(theta6,t_plot), subplot=2)\n",
    "scatter!(t, y, subplot=3,markersize = 2)\n",
    "plot!(t_plot, polyeval(theta10,t_plot), subplot=3)\n",
    "scatter!(t, y, subplot=4, markersize = 2)\n",
    "plot!(t_plot, polyeval(theta15,t_plot), subplot=4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Figure 13.4** Least squares polynomial fits of degree $1, 6, 10$, and $15$ to $100$\n",
    "points\n",
    "\n",
    "**Piecewise-linear fit.** In the following code least squares is used to fit a piecewise-linear function to $100$ points. It produces Figure 13.5, which is similar to Figure [13.8](https://web.stanford.edu/~boyd/vmls/vmls.pdf#figure.13.8) in VMLS."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
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     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# generate random data\n",
    "m = 100;\n",
    "x = -2 .+ 4*rand(m,1);\n",
    "y = 1 .+ 2*(x.-1) - 3*max.(x.+1,0) + 4*max.(x.-1,0) + 0.3*randn(m,1);\n",
    "# least squares fitting\n",
    "theta = [ ones(m) x max.(x.+1,0) max.(x.-1,0) ] \\ y;\n",
    "# plot result\n",
    "using Plots\n",
    "t = [-2.1, -1, 1, 2.1];\n",
    "yhat = theta[1] .+ theta[2]*t + theta[3]*max.(t.+1,0) +\n",
    "theta[4]*max.(t.-1,0);\n",
    "scatter(x, y, legend=false)\n",
    "plot!(t, yhat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Figure 13.5** Piecewise-linear fit to $100$ points.\n",
    "\n",
    "**House price regression.** We calculate the simple regression model for predicting\n",
    "house sales price from area and number of bedrooms, using the data of $774$ house\n",
    "sales in Sacramento."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{Float64,1}:\n",
       "  54.40167360394445\n",
       " 148.7250726002902 \n",
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   ],
   "source": [
    "D = house_sales_data(); # creates 3 vectors: area, beds, price\n",
    "area = D[\"area\"];\n",
    "beds = D[\"beds\"];\n",
    "price = D[\"price\"];\n",
    "m = length(price);\n",
    "A = [ ones(m) area beds ];\n",
    "x = A \\ price"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
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     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "rms_error = rms(price - A*x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
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     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "std_prices = stdev(price)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Auto-regressive time series model.** In the following Julia code we fit an auto-\n",
    "regressive model to the temperature time series discussed on page [259](https://web.stanford.edu/~boyd/vmls/vmls.pdf#section*.304) of VMLS. In Figure 13.6 we compare the first five days of the model predictions with the data. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "744"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = temperature_data(); # import time series of temperatures t\n",
    "N = length(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.05055928562933"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stdev(t) # Standard deviation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
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     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "rms(t[2:end] - t[1:end-1]) # RMS error for simple predictor zhat_{t+1} = z_t"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rms(t[25:end] - t[1:end-24]) # RMS error for simple predictor zhat_{t+1} = z_{t-23}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
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     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "M = 8 # Least squares fit of AR predictor with memory 8\n",
    "y = t[M+1:end];\n",
    "A = hcat( [ t[i:i+N-M-1] for i = M:-1:1]...);\n",
    "theta = A \\ y;\n",
    "ypred = A*theta;\n",
    "# RMS error of LS AR fit\n",
    "rms(ypred - y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Plot first five days\n",
    "using Plots\n",
    "Nplot = 24*5\n",
    "scatter(1:Nplot, t[1:Nplot], legend =false)\n",
    "plot!(M+1:Nplot, ypred[1:Nplot-M])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Figure 13.6** Hourly temperature at Los Angeles International Airport between 12:53AM on May 1, 2016, and 11:53PM on May 5, 2016, shown as circles. The solid line is the prediction of an auto-regressive model with eight coefficients."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 13.2 Validation\n",
    "**Polynomial approximation.** We return to the polynomial fitting example of page [100](#page_100). We continue with the data vectors `t` and `y` in the code on page [100](#page_100) as the training set, and generate a test set of $100$ randomly chosen points generated by the same method as used for the training set. We then fit polynomials of degree $0,. . . , 20$ (i.e., with $p = 1, . . . , 21$ coefficients) and compute the RMS errors on the training set and the test set. This produces a figure similar to Figure [13.11](https://web.stanford.edu/~boyd/vmls/vmls.pdf#figure.13.11) in VMLS, shown here as Figure 13.7. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
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     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Generate the test set.\n",
    "m = 100;\n",
    "t = -1 .+ 2*rand(m,1);\n",
    "\n",
    "y = t.^3 - t + 0.4 ./ (1 .+ 25*t.^2) + 0.10*randn(m,1);\n",
    "N = length(t)\n",
    "t_test = -1 .+ 2*rand(m,1);\n",
    "\n",
    "y_test = t_test.^3 - t_test + 0.4 ./ (1 .+ 25*t_test.^2) + 0.10*randn(m,1);\n",
    "error_train = zeros(21);\n",
    "error_test = zeros(21);\n",
    "\n",
    "for p = 1:21\n",
    "    A = vandermonde(t,p)\n",
    "    theta = A \\ y\n",
    "    error_train[p] = norm(A*theta - y) / norm(y)\n",
    "    error_test[p] = norm( vandermonde(t_test, p) * theta - y_test) / norm(y_test);\n",
    "end\n",
    "using Plots\n",
    "plot(0:20, error_train, label = \"Train\", marker = :circle)\n",
    "plot!(0:20, error_test, label = \"Test\", marker = :square)\n",
    "plot!(xlabel=\"Degree\", ylabel = \"Relative RMS error\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Figure 13.7** RMS error versus polynomial degree for the fitting example in\n",
    "figure 13.4.\n",
    "\n",
    "**House price regression model.** On page 13.1 we used a data set of $774$ house sales data to fit a simple regression model\n",
    "$$\n",
    "ŷ = v + β_1x_1 + β_2x_2,\n",
    "$$ where $ŷ$ is the predicted sales price, $x_1$ is the area, and $x_2$ is the number of bedrooms. Here we apply cross-validation to assess the generalization ability of the simple model. We use five folds, four of size $155$ ($N$fold in the code below) and one of size $154$. To choose the five folds, we create a random permutation of the indices $1, . . . , 774$. (We do this by calling the randperm function in the `Random` package.) We choose the data points indexed by the first $155$ elements in the permuted list as fold 1, the next $155$ as fold $2$, et cetera. The output of the following code outputs is similar to Table [13.1](https://web.stanford.edu/~boyd/vmls/vmls.pdf#table.13.1) in VMLS (with different numbers because of the random choice of folds). \n",
    "\n",
    "<a id='page_106'></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×3 Array{Float64,2}:\n",
       " 54.5823  149.899  -18.9994\n",
       " 51.0057  146.893  -17.2246\n",
       " 54.8184  149.087  -20.0823\n",
       " 54.539   146.728  -17.7843\n",
       " 57.3916  150.641  -20.0933"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = house_sales_data();\n",
    "price = D[\"price\"]; area = D[\"area\"]; beds = D[\"beds\"];\n",
    "N = length(price);\n",
    "X = [ ones(N) area beds ];\n",
    "nfold = div(N,5); # size of first four folds\n",
    "import Random\n",
    "I = Random.randperm(N); # random permutation of numbers 1...N\n",
    "coeff = zeros(5,3); errors = zeros(5,2);\n",
    "rms_train = zeros(0)\n",
    "rms_test = zeros(0)\n",
    "for k = 1:5\n",
    "    if k == 1\n",
    "        Itrain = I[nfold+1:end];\n",
    "        Itest = I[1:nfold];\n",
    "        elseif k == 5\n",
    "        Itrain = I[1:4*nfold];\n",
    "        Itest = I[4*nfold+1:end];\n",
    "    else\n",
    "        Itrain = I[ [1:(k-1)*nfold ; k*nfold+1 : N]]\n",
    "        Itest = I[ [(k-1)*nfold+1 ; k*nfold ]];\n",
    "        end;\n",
    "    Ntrain = length(Itrain)\n",
    "    Ntest = length(Itest)\n",
    "    theta = X[Itrain,:] \\ price[Itrain];\n",
    "    coeff[k,:] = theta;\n",
    "    append!(rms_train, rms(X[Itrain,:] * theta - price[Itrain]))\n",
    "    append!(rms_test, rms(X[Itest,:] * theta - price[Itest]))\n",
    "    end;\n",
    "coeff # 3 coefficients for the five folds"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×2 Array{Float64,2}:\n",
       " 76.5994  67.4516\n",
       " 74.2508  28.4643\n",
       " 75.2113  58.7357\n",
       " 73.4841  44.6074\n",
       " 74.325   77.3214"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[rms_train rms_test] # RMS errors for five folds"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Validating time series predictions.** In the next example, we return to the $AR$\n",
    "model of hourly temperatures at $LAX$. We divide the time series in a training set\n",
    "of $24$ days and a test set of $7$ days. We fit the $AR$ model to the training set and\n",
    "calculate the $RMS$ prediction errors on the training and test sets. Figure 13.8\n",
    "shows the model predictions and the the data for the first five days of the test set."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0253577259862334"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = temperature_data();\n",
    "N = length(t);\n",
    "# use first 24 days as training set\n",
    "Ntrain = 24 * 24; t_train = t[1:Ntrain];\n",
    "# use the rest as test set\n",
    "Ntest = N-Ntrain; t_test = t[Ntrain+1:end];\n",
    "# Least squares fit of AR predictor with memory 8\n",
    "m = Ntrain - M;\n",
    "y = t_train[M+1:M+m];\n",
    "A = hcat( [t_train[i:i+m-1] for i=M:-1:1]...);\n",
    "coeff = A \\ y;\n",
    "rms_train = rms(A*coeff-y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9755113632200977"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ytest = t_test[M+1:end];\n",
    "mtest = length(ytest);\n",
    "ypred = hcat( [t_test[i:i+mtest-1] for i=M:-1:1]...) * coeff;\n",
    "rms_test = rms(ypred - ytest)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "Nplot = 24*5\n",
    "scatter(1:Nplot, t_test[1:Nplot], legend=false)\n",
    "plot!(M+1:Nplot, ypred[1:Nplot-M])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Figure 13.8** Hourly temperature at Los Angeles International Airport be-\n",
    "tween $12:53AM$ on $May 25$, $2016$, and $11:53PM$ on $May 29$, $2016$, shown\n",
    "as circles. The solid line is the prediction of an auto-regressive model with\n",
    "eight coefficients, developed using training data from $May 1$ to $May 24$.\n",
    "\n",
    "### 13.3 Feature engineering\n",
    "Next we compute the more complicated house price regression model of §[13.3.5](https://web.stanford.edu/~boyd/vmls/vmls.pdf#subsection.13.3.5) of VMLS. The data are imported via the function house_sales_data, which returns a dictionary containing the following five vectors of length `774`: \n",
    "* price: price in thousand dollars,\n",
    "* area: house area in $1000$ square feet,\n",
    "* beds: number of bedrooms,\n",
    "* condo: `1` if a condo, `0` otherwise,\n",
    "* location: a number from `1–4`, for the four sets of ZIP codes in Table [13.4](https://web.stanford.edu/~boyd/vmls/vmls.pdf#table.13.4) of VMLS.\n",
    "\n",
    "The code computes the model and makes a scatter plot of actual and predicted\n",
    "prices (Figure 13.9). Note that the last three columns of the matrix `X` contain\n",
    "$Boolean$ variables (`true` or `false`). We rely on the fact that Julia treats this as\n",
    "integers `1` and `0`. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8-element Array{Float64,1}:\n",
       "  115.61682367025571 \n",
       "  175.4131406399585  \n",
       "  -42.74776796782588 \n",
       "  -17.87835523524344 \n",
       "  -19.04472565054657 \n",
       " -100.91050308608219 \n",
       " -108.79112222093232 \n",
       "  -24.765247347062342"
      ]
     },
     "execution_count": 130,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = house_sales_data();\n",
    "price = D[\"price\"];\n",
    "area = D[\"area\"];\n",
    "beds = D[\"beds\"];\n",
    "condo = D[\"condo\"];\n",
    "location = D[\"location\"];\n",
    "N = length(price);\n",
    "X = hcat(ones(N), area, max.(area.-1.5, 0), beds, condo, location .== 2, location .== 3, location .== 4 );\n",
    "theta = X \\ price"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "68.34428699036884"
      ]
     },
     "execution_count": 131,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rms(X*theta - price) # RMS prediction error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 132,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "scatter(price, X*theta, lims = (0,800))\n",
    "plot!([0, 800], [0, 800], linestyle = :dash, legend = false)\n",
    "plot!(xlims = (0,800), ylims = (0,800), size = (500,500))\n",
    "plot!(xlabel = \"Actual price\", ylabel = \"Predicted price\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Figure 13.9** Scatter plot of actual and predicted prices for a model with eight\n",
    "parameters.\n",
    "    \n",
    "We finish by a cross-validation of this method. We follow the same approach as for\n",
    "the simple regression model on page [106](#page_106), using five randomly chosen folds. The\n",
    "code shows the eight coefficients, and the RMS training and test errors for each\n",
    "fold."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8×5 Array{Float64,2}:\n",
       "  113.126    98.6852    122.382    126.859    117.017 \n",
       "  172.769   182.61      180.0      165.741    176.493 \n",
       "  -43.7341  -50.329     -46.118    -27.4887   -46.6907\n",
       "  -13.6015  -20.1239    -19.931    -18.1421   -17.9758\n",
       "  -16.7486  -22.417     -15.0073   -21.8539   -19.2651\n",
       " -108.818   -85.7231   -109.74     -99.402   -100.107 \n",
       " -117.524   -92.4818   -112.467   -106.409   -114.247 \n",
       "  -34.8623   -9.94509   -33.223    -21.7424   -22.9404"
      ]
     },
     "execution_count": 134,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = length(price);\n",
    "nfold = div(N,5);\n",
    "import Random;\n",
    "I = Random.randperm(N);\n",
    "models = zeros(8,5); # store 8 coefficients for the 5 models\n",
    "errors = zeros(2,5); # prediction errors\n",
    "for k = 1:5\n",
    "    if k == 1\n",
    "        Itrain = I[nfold+1:end];\n",
    "        Itest = I[1:nfold];\n",
    "        elseif k == 5\n",
    "        Itrain = I[1:4*nfold];\n",
    "        Itest = I[4*nfold+1:end];\n",
    "    else\n",
    "        Itrain = I[ [1:(k-1)*nfold ; k*nfold+1 : N]]\n",
    "        Itest = I[ [(k-1)*nfold+1 ; k*nfold ]];\n",
    "        end;\n",
    "    Ntrain = length(Itrain)\n",
    "    Ntest = length(Itest)\n",
    "    theta = X[Itrain,:] \\ price[Itrain];\n",
    "    errors[1,k] = rms(X[Itrain,:] * theta - price[Itrain]);\n",
    "    errors[2,k] = rms(X[Itest,:] * theta - price[Itest]);\n",
    "    models[:,k] = theta;\n",
    "end;\n",
    "# display the eight coefficients for each of the 5 folds \n",
    "models"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×5 Array{Float64,2}:\n",
       " 68.0707  69.3719   68.2224   66.2214  69.44  \n",
       " 69.761    5.58659  63.6452  126.556   64.3574"
      ]
     },
     "execution_count": 135,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#display training errors (1st row) and test errors (2nd) row\n",
    "errors"
   ]
  }
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